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Existence results for some optimal partition problems. (English) Zbl 0915.49006

If \({\mathcal A}(\Omega)\) is a suitably defined class of admissible subdomains of a domain \(\Omega\) and \(F\) is a functional defined in \({\mathcal A}(\Omega)\), the problem \[ \min \{F(A);\;A \in {\mathcal A}(\Omega)\} \] is a shape optimization problem. A generalization is the optimal partition problem, \[ \min \{F(A_1, A_2, \dots, A_k); \;A_j \in{\mathcal A}(\Omega), \;A_i \cap A_j = \emptyset \;(i \neq j)\} , \] where \(k\) is a positive integer and \(F\) is a functional defined in \({\mathcal A}(\Omega)^k\). The object is to show that, for functionals satisfying a suitable monotonicity assumption, a classical (as opposed to relaxed) solution of this problem exists. Motivation comes from the problem of minimizing \[ J(A, B) = \lambda_1(A) + \lambda_1(B) \] with \(A, B \subseteq \Omega\), \(A \cap B = \emptyset\), \(\lambda_1\) the first Dirichlet eigenvalue of \(-\Delta\) in each domain. The authors put forth (but do not settle) the conjecture that the minimum is \(2\lambda_2(\Omega)\), \(A, B\) the two nodal domains of the second Dirichlet eigenfunction \(\varphi_2(x)\) in \(\Omega\) with eigenvalue \(\lambda_2(\Omega)\). This conjecture is shown to be related to another (due to L. E. Payne) to the effect that \(\varphi_2(x)\) cannot have closed nodal lines.

MSC:

49J45 Methods involving semicontinuity and convergence; relaxation
49Q10 Optimization of shapes other than minimal surfaces
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